The classical Faber–Krahn inequality asserts that balls (uniquely) minimize the first eigenvalue of the Dirichlet Laplacian among sets with given volume. In this article we prove a sharp quantitative enhancement of this result, thus confirming a conjecture by Nadirashvili and by Bhattacharya and Weitsman. More generally, the result applies to every optimal Poincaré–Sobolev constant for the embeddings .
"Faber–Krahn inequalities in sharp quantitative form." Duke Math. J. 164 (9) 1777 - 1831, 15 June 2015. https://doi.org/10.1215/00127094-3120167